Mercurial > octave
view scripts/sparse/gmres.m @ 29949:f254c302bb9c
remove JIT compiler from Octave sources
As stated in the NEWS file entry added with this changeset, no one
has ever seriously taken on further development of the JIT compiler in
Octave since it was first added as part of a Google Summer of Code
project in 2012 and it still does nothing significant. It is out of
date with the default interpreter that walks the parse tree. Even
though we have fixed the configure script to disable it by default,
people still ask questions about how to build it, but it doesn’t seem
that they are doing that to work on it but because they think it will
make Octave code run faster (it never did, except for some extremely
simple bits of code as examples for demonstration purposes only).
* NEWS: Note change.
* configure.ac, acinclude.m4: Eliminate checks and macros related to
the JIT compiler and LLVM.
* basics.txi, install.txi, octave.texi, vectorize.txi: Remove mention
of JIT compiler and LLVM.
* jit-ir.cc, jit-ir.h, jit-typeinfo.cc, jit-typeinfo.h, jit-util.cc,
jit-util.h, pt-jit.cc, pt-jit.h: Delete.
* libinterp/parse-tree/module.mk: Update.
* Array-jit.cc: Delete.
* libinterp/template-inst/module.mk: Update.
* test/jit.tst: Delete.
* test/module.mk: Update.
* interpreter.cc (interpreter::interpreter): Don't check options for
debug_jit or jit_compiler.
* toplev.cc (F__octave_config_info__): Remove JIT compiler and LLVM
info from struct.
* ov-base.h (octave_base_value::grab, octave_base_value::release):
Delete.
* ov-builtin.h, ov-builtin.cc (octave_builtin::to_jit,
octave_builtin::stash_jit): Delete.
(octave_builtin::m_jtype): Delete data member and all uses.
* ov-usr-fcn.h, ov-usr-fcn.cc (octave_user_function::m_jit_info):
Delete data member and all uses.
(octave_user_function::get_info, octave_user_function::stash_info): Delete.
* options.h (DEBUG_JIT_OPTION, JIT_COMPILER_OPTION): Delete macro
definitions and all uses.
* octave.h, octave.cc (cmdline_options::cmdline_options): Don't handle
DEBUG_JIT_OPTION, JIT_COMPILER_OPTION): Delete.
(cmdline_options::debug_jit, cmdline_options::jit_compiler): Delete
functions and all uses.
(cmdline_options::m_debug_jit, cmdline_options::m_jit_compiler): Delete
data members and all uses.
(octave_getopt_options long_opts): Remove "debug-jit" and
"jit-compiler" from the list.
* pt-eval.cc (tree_evaluator::visit_simple_for_command,
tree_evaluator::visit_complex_for_command,
tree_evaluator::visit_while_command,
tree_evaluator::execute_user_function): Eliminate JIT compiler code.
* pt-loop.h, pt-loop.cc (tree_while_command::get_info,
tree_while_command::stash_info, tree_simple_for_command::get_info,
tree_simple_for_command::stash_info): Delete functions and all uses.
(tree_while_command::m_compiled, tree_simple_for_command::m_compiled):
Delete member variable and all uses.
* usage.h (usage_string, octave_print_verbose_usage_and_exit): Remove
[--debug-jit] and [--jit-compiler] from the message.
* Array.h (Array<T>::Array): Remove constructor that was only intended
to be used by the JIT compiler.
(Array<T>::jit_ref_count, Array<T>::jit_slice_data,
Array<T>::jit_dimensions, Array<T>::jit_array_rep): Delete.
* Marray.h (MArray<T>::MArray): Remove constructor that was only
intended to be used by the JIT compiler.
* NDArray.h (NDArray::NDarray): Remove constructor that was only
intended to be used by the JIT compiler.
* dim-vector.h (dim_vector::to_jit): Delete.
(dim_vector::dim_vector): Remove constructor that was only intended to
be used by the JIT compiler.
* codeql-analysis.yaml, make.yaml: Don't require llvm-dev.
* subst-config-vals.in.sh, subst-cross-config-vals.in.sh: Don't
substitute OCTAVE_CONF_LLVM_CPPFLAGS, OCTAVE_CONF_LLVM_LDFLAGS, or
OCTAVE_CONF_LLVM_LIBS.
* Doxyfile.in: Don't define HAVE_LLVM.
* aspell-octave.en.pws: Eliminate jit, JIT, and LLVM from the list of
spelling exceptions.
* build-env.h, build-env.in.cc (LLVM_CPPFLAGS, LLVM_LDFLAGS,
LLVM_LIBS): Delete variables and all uses.
* libinterp/corefcn/module.mk (%canon_reldir%_libcorefcn_la_CPPFLAGS):
Remove $(LLVM_CPPFLAGS) from the list.
* libinterp/parse-tree/module.mk (%canon_reldir%_libparse_tree_la_CPPFLAGS):
Remove $(LLVM_CPPFLAGS) from the list.
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Tue, 10 Aug 2021 16:42:29 -0400 |
parents | b2455f0a8297 |
children | 796f54d4ddbf |
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######################################################################## ## ## Copyright (C) 2009-2021 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or <https://octave.org/copyright/>. ## ## This file is part of Octave. ## ## Octave is free software: you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <https://www.gnu.org/licenses/>. ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {@var{x} =} gmres (@var{A}, @var{b}, @var{restart}, @var{tol}, @var{maxit}, @var{M1}, @var{M2}, @var{x0}, @dots{}) ## @deftypefnx {} {@var{x} =} gmres (@var{A}, @var{b}, @var{restart}, @var{tol}, @var{maxit}, @var{M}, [], @var{x0}, @dots{}) ## @deftypefnx {} {[@var{x}, @var{flag}, @var{relres}, @var{iter}, @var{resvec}] =} gmres (@var{A}, @var{b}, @dots{}) ## Solve @code{A x = b} using the Preconditioned GMRES iterative method with ## restart, a.k.a. PGMRES(restart). ## ## The input arguments are: ## ## @itemize @minus ## ## @item @var{A} is the matrix of the linear system and it must be square. ## @var{A} can be passed as a matrix, function handle, or inline ## function @code{Afun} such that @code{Afun(x) = A * x}. Additional ## parameters to @code{Afun} are passed after @var{x0}. ## ## @item @var{b} is the right hand side vector. It must be a column vector ## with the same numbers of rows as @var{A}. ## ## @item @var{restart} is the number of iterations before that the ## method restarts. If it is [] or N = numel (b), then the restart ## is not applied. ## ## @item @var{tol} is the required relative tolerance for the ## preconditioned residual error, ## @code{inv (@var{M}) * (@var{b} - @var{a} * @var{x})}. The iteration ## stops if @code{norm (inv (@var{M}) * (@var{b} - @var{a} * @var{x})) ## @leq{} @var{tol} * norm (inv (@var{M}) * @var{B})}. If @var{tol} is ## omitted or empty, then a tolerance of 1e-6 is used. ## ## @item @var{maxit} is the maximum number of outer iterations, if not given or ## set to [], then the default value @code{min (10, @var{N} / @var{restart})} ## is used. ## Note that, if @var{restart} is empty, then @var{maxit} is the maximum number ## of iterations. If @var{restart} and @var{maxit} are not empty, then ## the maximum number of iterations is @code{@var{restart} * @var{maxit}}. ## If both @var{restart} and @var{maxit} are empty, then the maximum ## number of iterations is set to @code{min (10, @var{N})}. ## ## @item @var{M1}, @var{M2} are the preconditioners. The preconditioner ## @var{M} is given as @code{M = M1 * M2}. Both @var{M1} and @var{M2} can ## be passed as a matrix, function handle, or inline function @code{g} such ## that @code{g(x) = M1 \ x} or @code{g(x) = M2 \ x}. If @var{M1} is [] or not ## given, then the preconditioner is not applied. ## The technique used is the left-preconditioning, i.e., it is solved ## @code{inv(@var{M}) * @var{A} * @var{x} = inv(@var{M}) * @var{b}} instead of ## @code{@var{A} * @var{x} = @var{b}}. ## ## @item @var{x0} is the initial guess, ## if not given or set to [], then the default value ## @code{zeros (size (@var{b}))} is used. ## ## @end itemize ## ## The arguments which follow @var{x0} are treated as parameters, and passed in ## a proper way to any of the functions (@var{A} or @var{M} or ## @var{M1} or @var{M2}) which are passed to @code{gmres}. ## ## The outputs are: ## ## @itemize @minus ## ## @item @var{x} the computed approximation. If the method does not ## converge, then it is the iterated with minimum residual. ## ## @item @var{flag} indicates the exit status: ## ## @table @asis ## @item 0 : iteration converged to within the specified tolerance ## ## @item 1 : maximum number of iterations exceeded ## ## @item 2 : the preconditioner matrix is singular ## ## @item 3 : algorithm reached stagnation (the relative difference between two ## consecutive iterations is less than eps) ## @end table ## ## @item @var{relres} is the value of the relative preconditioned ## residual of the approximation @var{x}. ## ## @item @var{iter} is a vector containing the number of outer iterations and ## inner iterations performed to compute @var{x}. That is: ## ## @itemize ## @item @var{iter(1)}: number of outer iterations, i.e., how many ## times the method restarted. (if @var{restart} is empty or @var{N}, ## then it is 1, if not 1 @leq{} @var{iter(1)} @leq{} @var{maxit}). ## ## @item @var{iter(2)}: the number of iterations performed before the ## restart, i.e., the method restarts when ## @code{@var{iter(2)} = @var{restart}}. If @var{restart} is empty or ## @var{N}, then 1 @leq{} @var{iter(2)} @leq{} @var{maxit}. ## @end itemize ## ## To be more clear, the approximation @var{x} is computed at the iteration ## @code{(@var{iter(1)} - 1) * @var{restart} + @var{iter(2)}}. ## Since the output @var{x} corresponds to the minimal preconditioned ## residual solution, the total number of iterations that ## the method performed is given by @code{length (resvec) - 1}. ## ## @item @var{resvec} is a vector containing the preconditioned ## relative residual at each iteration, including the 0-th iteration ## @code{norm (@var{A} * @var{x0} - @var{b})}. ## @end itemize ## ## Let us consider a trivial problem with a tridiagonal matrix ## ## @example ## @group ## n = 20; ## A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ... ## toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ... ## sparse (1, 2, 1, 1, n) * n / 2); ## b = A * ones (n, 1); ## restart = 5; ## [M1, M2] = ilu (A); # in this tridiag case, it corresponds to lu (A) ## M = M1 * M2; ## Afun = @@(x) A * x; ## Mfun = @@(x) M \ x; ## M1fun = @@(x) M1 \ x; ## M2fun = @@(x) M2 \ x; ## @end group ## @end example ## ## @sc{Example 1:} simplest usage of @code{gmres} ## ## @example ## x = gmres (A, b, [], [], n) ## @end example ## ## @sc{Example 2:} @code{gmres} with a function which computes ## @code{@var{A} * @var{x}} ## ## @example ## x = gmres (Afun, b, [], [], n) ## @end example ## ## @sc{Example 3:} usage of @code{gmres} with the restart ## ## @example ## x = gmres (A, b, restart); ## @end example ## ## @sc{Example 4:} @code{gmres} with a preconditioner matrix @var{M} ## with and without restart ## ## @example ## @group ## x = gmres (A, b, [], 1e-06, n, M) ## x = gmres (A, b, restart, 1e-06, n, M) ## @end group ## @end example ## ## @sc{Example 5:} @code{gmres} with a function as preconditioner ## ## @example ## x = gmres (Afun, b, [], 1e-6, n, Mfun) ## @end example ## ## @sc{Example 6:} @code{gmres} with preconditioner matrices @var{M1} ## and @var{M2} ## ## @example ## x = gmres (A, b, [], 1e-6, n, M1, M2) ## @end example ## ## @sc{Example 7:} @code{gmres} with functions as preconditioners ## ## @example ## x = gmres (Afun, b, 1e-6, n, M1fun, M2fun) ## @end example ## ## @sc{Example 8:} @code{gmres} with as input a function requiring an argument ## ## @example ## @group ## function y = Ap (A, x, p) # compute A^p * x ## y = x; ## for i = 1:p ## y = A * y; ## endfor ## endfunction ## Apfun = @@(x, p) Ap (A, x, p); ## x = gmres (Apfun, b, [], [], [], [], [], [], 2); ## @end group ## @end example ## ## @sc{Example 9:} explicit example to show that @code{gmres} uses a ## left preconditioner ## ## @example ## @group ## [M1, M2] = ilu (A + 0.1 * eye (n)); # factorization of A perturbed ## M = M1 * M2; ## ## ## reference solution computed by gmres after two iterations ## [x_ref, fl] = gmres (A, b, [], [], 1, M) ## ## ## left preconditioning ## [x, fl] = gmres (M \ A, M \ b, [], [], 1) ## x # compare x and x_ref ## ## @end group ## @end example ## ## Reference: ## ## @nospell{Y. Saad}, @cite{Iterative Methods for Sparse Linear ## Systems}, Second edition, 2003, SIAM ## ## @seealso{bicg, bicgstab, cgs, pcg, pcr, qmr, tfqmr} ## @end deftypefn function [x_min, flag, relres, it, resvec] = ... gmres (A, b, restart = [], tol = [], maxit = [], M1 = [], M2 = [], x0 = [], varargin) if (strcmp (class (A), "single") || strcmp (class (b), "single")) class_name = "single"; else class_name = "double"; endif [Afun, M1fun, M2fun] = __alltohandles__ (A, b, M1, M2, "gmres"); ## Check if the inputs are empty, and in case set them [tol, x0] = __default__input__ ({1e-06, zeros(size (b))}, tol, x0); empty_restart = isempty (restart); empty_maxit = isempty (maxit); size_b = rows (b); if (tol >= 1) warning ("Input tol is bigger than 1. \n Try to use a smaller tolerance."); elseif (tol <= eps / 2) warning ("Input tol may not be achievable by gmres. \n Try to use a bigger tolerance."); endif ## This big "if block" is to set maxit and restart in the proper way if ((empty_restart) && (empty_maxit)) restart = size_b; maxit = 1; max_iter_number = min (size_b, 10); elseif (restart <= 0) || (maxit <= 0) error ("gmres: MAXIT and RESTART must be positive integers"); elseif (restart < size_b) && (empty_maxit) maxit = min (size_b / restart, 10); max_iter_number = maxit * restart; elseif (restart == size_b) && (empty_maxit) maxit = 1; max_iter_number = min (size_b, 10); elseif (restart > size_b) && (empty_maxit) warning ("RESTART is %d but it should be bounded by SIZE(A,2).\n Setting restart to %d. \n", restart, size_b); restart = size_b; maxit = 1; max_iter_number = restart; elseif (empty_restart) && (maxit <= size_b) restart = size_b; max_iter_number = maxit; elseif (empty_restart) && (maxit > size_b) warning ("MAXIT is %d but it should be bounded by SIZE(A,2). \n Setting MAXIT to %d", maxit, size_b); restart = size_b; maxit = size_b; max_iter_number = size_b; elseif (restart > size_b) && (! empty_maxit) warning ("RESTART is %d but it should be bounded by SIZE(A,2).\n Setting restart to %d. \n", restart, size_b); restart = size_b; max_iter_number = restart * maxit; elseif (restart == size_b) && (maxit <= size_b) max_iter_number = maxit; else max_iter_number = restart*maxit; endif prec_b_norm = norm (b, 2); if (prec_b_norm == 0) if (nargout < 2) printf ("The right hand side vector is all zero so gmres\nreturned an all zero solution without iterating.\n") endif x_min = b; flag = 0; relres = 0; resvec = 0; it = [0, 0]; return; endif ## gmres: function handle case x_old = x_pr = x_min = x = x0; B = zeros (restart + 1, 1); V = zeros (rows (x), restart, class_name); H = zeros (restart + 1, restart); iter = 1; # total number of iterations iter_min = 0; # iteration with minimum residual outer_it = 1; # number of outer iterations restart_it = 1; # number of inner iterations it = zeros (1, 2); resvec = zeros (max_iter_number + 1, 1); flag = 1; # Default flag is maximum # of iterations exceeded ## begin loop u = feval (Afun, x_old, varargin{:}); try warning ("error", "Octave:singular-matrix", "local"); prec_res = feval (M1fun, b - u, varargin{:}); # M1*(b-u) prec_res = feval (M2fun, prec_res, varargin{:}); presn = norm (prec_res, 2); resvec(1) = presn; z = feval (M1fun, b, varargin{:}); z = feval (M2fun, z, varargin{:}); prec_b_norm = norm (z, 2); B (1) = presn; V(:, 1) = prec_res / presn; catch flag = 2; end_try_catch while (flag != 2) && (iter <= max_iter_number) && ... (presn > tol * prec_b_norm) ## restart if (restart_it > restart) restart_it = 1; outer_it += 1; x_old = x; u = feval (Afun, x_old, varargin{:}); prec_res = feval (M1fun, b - u, varargin{:}); prec_res = feval (M2fun, prec_res, varargin{:}); presn = norm (prec_res, 2); B(1) = presn; H(:) = 0; V(:, 1) = prec_res / presn; endif ## basic iteration u = feval (Afun, V(:, restart_it), varargin{:}); tmp = feval (M1fun, u, varargin{:}); tmp = feval (M2fun, tmp, varargin{:}); [V(:,restart_it + 1), H(1:restart_it + 1, restart_it)] = ... mgorth (tmp, V(:,1:restart_it)); Y = (H(1:restart_it + 1, 1:restart_it) \ B(1:restart_it + 1)); little_res = B(1:restart_it + 1) - ... H(1:restart_it + 1, 1:restart_it) * Y(1:restart_it); presn = norm (little_res, 2); x = x_old + V(:, 1:restart_it) * Y(1:restart_it); resvec(iter + 1) = presn; if (norm (x - x_pr) <= eps*norm (x)) flag = 3; # Stagnation: little change between iterations break; endif if (resvec (iter + 1) <= resvec (iter_min + 1)) x_min = x; iter_min = iter; it = [outer_it, restart_it]; endif x_pr = x; restart_it += 1; iter += 1; endwhile if (flag == 2) resvec = norm (b); relres = 1; else resvec = resvec (1:iter); relres = resvec (iter) / prec_b_norm; endif if ((relres <= tol) && (flag == 1)) flag = 0; # Converged to solution within tolerance endif if ((nargout < 2) && (restart != size_b)) # restart applied switch (flag) case {0} # gmres converged printf ("gmres (%d) converged at outer iteration %d (inner iteration %d) ",restart, it (1), it (2)); printf ("to a solution with relative residual %d \n", relres); case {1} # max number of iteration reached printf ("gmres (%d) stopped at outer iteration %d (inner iteration %d) ", restart, outer_it, restart_it-1); printf ("without converging to the desired tolerance %d ", tol); printf ("because the maximum number of iterations was reached \n"); printf ("The iterated returned (number %d(%d)) ", it(1), it(2)); printf ("has relative residual %d \n", relres); case {2} # preconditioner singular printf ("gmres (%d) stopped at outer iteration %d (inner iteration %d) ",restart, outer_it, restart_it-1); printf ("without converging to the desired tolerance %d ", tol); printf ("because the preconditioner matrix is singular \n"); printf ("The iterated returned (number %d(%d)) ", it(1), it(2)); printf ("has relative residual %d \n", relres); case {3} # stagnation printf ("gmres (%d) stopped at outer iteration %d (inner iteration %d) ", restart, outer_it, restart_it - 1); printf ("without converging to the desired tolerance %d", tol); printf ("because it stagnates. \n"); printf ("The iterated returned (number %d(%d)) ", it(1), it(2)); printf ("has relative residual %d \n", relres); endswitch elseif ((nargout < 2) && (restart == size_b)) # no restart switch (flag) case {0} # gmres converged printf ("gmres converged at iteration %d ", it(2)); printf ("to a solution with relative residual %d \n", relres); case {1} # max number of iteration reached printf ("gmres stopped at iteration %d ", restart_it - 1); printf ("without converging to the desired tolerance %d ", tol); printf ("because the maximum number of iterations was reached \n"); printf ("The iterated returned (number %d) ", it(2)); printf ("has relative residual %d \n", relres); case {2} # preconditioner ill-conditioned printf ("gmres stopped at iteration %d ", restart_it - 1); printf ("without converging to the desired tolerance %d ", tol); printf ("because the preconditioner matrix is singular \n") printf ("The iterated returned (number %d) ", it (2)); printf ("has relative residual %d \n", relres); case {3} # stagnation printf ("gmres stopped at iteration %d ", restart_it - 1); printf ("without converging at the desired tolerance %d ", tol); printf ("because it stagnates\n"); printf ("The iterated returned (number %d) ", it(2)); printf ("has relative residual %d \n", relres); endswitch endif endfunction %!demo %! dim = 20; %! A = spdiags ([-ones(dim,1) 2*ones(dim,1) ones(dim,1)], [-1:1], dim, dim); %! b = ones (dim, 1); %! [x, flag, relres, iter, resvec] = ... %! gmres (A, b, 10, 1e-10, dim, @(x) x ./ diag (A), [], b) %!demo # simplest use %! n = 20; %! A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ... %! toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ... %! sparse (1, 2, 1, 1, n) * n / 2); %! b = A * ones (n, 1); %! restart = 5; %! [M1, M2] = ilu (A + 0.1 * eye (n)); %! M = M1 * M2; %! x = gmres (A, b, [], [], n); %! x = gmres (A, b, restart, [], n); # gmres with restart %! Afun = @(x) A * x; %! x = gmres (Afun, b, [], [], n); %! x = gmres (A, b, [], 1e-6, n, M); # gmres without restart %! x = gmres (A, b, [], 1e-6, n, M1, M2); %! Mfun = @(x) M \ x; %! x = gmres (Afun, b, [], 1e-6, n, Mfun); %! M1fun = @(x) M1 \ x; %! M2fun = @(x) M2 \ x; %! x = gmres (Afun, b, [], 1e-6, n, M1fun, M2fun); %! function y = Ap (A, x, p) # compute A^p * x %! y = x; %! for i = 1:p %! y = A * y; %! endfor %! endfunction %! Afun = @(x, p) Ap (A, x, p); %! x = gmres (Afun, b, [], [], n, [], [], [], 2); # solution of A^2 * x = b %!demo %! n = 10; %! A = toeplitz (sparse ([1, 1], [1, 2], [2, 1] * n ^ 2, 1, n)) + ... %! toeplitz (sparse (1, 2, -1, 1, n) * n / 2, ... %! sparse (1, 2, 1, 1, n) * n / 2); %! b = A * ones (n, 1); %! [M1, M2] = ilu (A + 0.1 * eye (n)); # factorization of A perturbed %! M = M1 * M2; %! %! ## reference solution computed by gmres after one iteration %! [x_ref, fl] = gmres (A, b, [], [], 1, M); %! x_ref %! %! ## left preconditioning %! [x, fl] = gmres (M \ A, M \ b, [], [], 1); %! x # compare x and x_ref %!test %! ## Check that all type of inputs work %! A = toeplitz (sparse ([2, 1, 0, 0, 0]), sparse ([2, -1, 0, 0, 0])); %! b = sum (A, 2); %! M1 = diag (sqrt (diag (A))); %! M2 = M1; %! Afun = @(z) A * z; %! M1_fun = @(z) M1 \ z; %! M2_fun = @(z) M2 \ z; %! [x, flag] = gmres (A, b); %! assert (flag, 0); %! [x, flag] = gmres (A, b, [], [], [], M1, M2); %! assert (flag, 0); %! [x, flag] = gmres (A, b, [], [], [], M1_fun, M2_fun); %! assert (flag, 0); %! [x, flag] = gmres (A, b, [], [], [], M1_fun, M2); %! assert (flag, 0); %! [x, flag] = gmres (A, b, [], [], [], M1, M2_fun); %! assert (flag, 0); %! [x, flag] = gmres (Afun, b); %! assert (flag, 0); %! [x, flag] = gmres (Afun, b, [],[],[], M1, M2); %! assert (flag, 0); %! [x, flag] = gmres (Afun, b, [],[],[], M1_fun, M2); %! assert (flag, 0); %! [x, flag] = gmres (Afun, b, [],[],[], M1, M2_fun); %! assert (flag, 0); %! [x, flag] = gmres (Afun, b, [],[],[], M1_fun, M2_fun); %! assert (flag, 0); %!test %! dim = 100; %! A = spdiags ([-ones(dim,1), 2*ones(dim,1), ones(dim,1)], [-1:1], dim, dim); %! b = ones (dim, 1); %! [x, flag] = gmres (A, b, 10, 1e-10, dim, @(x) x ./ diag (A), [], b); %! assert (x, A\b, 1e-9*norm (x, Inf)); %! [x, flag] = gmres (A, b, dim, 1e-10, 1e4, @(x) diag (diag (A)) \ x, [], b); %! assert (x, A\b, 1e-7*norm (x, Inf)); %!test %! dim = 100; %! A = spdiags ([[1./(2:2:2*(dim-1)) 0]; 1./(1:2:2*dim-1); ... %! [0 1./(2:2:2*(dim-1))]]', -1:1, dim, dim); %! A = A'*A; %! b = rand (dim, 1); %! [x, resvec] = gmres (@(x) A*x, b, dim, 1e-10, dim, ... %! @(x) x./diag (A), [], []); %! assert (x, A\b, 1e-9*norm (x, Inf)); %! [x, flag] = gmres (@(x) A*x, b, dim, 1e-10, 1e5, ... %! @(x) diag (diag (A)) \ x, [], []); %! assert (x, A\b, 1e-9*norm (x, Inf)); %! [x, flag] = gmres (@(x) A*x, b, dim, 1e-10, 1e5, ... %! @(x) x ./ diag (A), [], []); %! assert (x, A\b, 1e-7*norm (x, Inf)); %!test %! ## gmres solves complex linear systems %! A = toeplitz (sparse ([2, 1, 0, 0, 0]), sparse ([2, -1, 0, 0, 0])) + ... %! 1i * toeplitz (sparse ([2, 1, 0, 0, 0]), sparse ([2, -1, 0, 0, 0])); %! b = sum (A, 2); %! [x, flag] = gmres(A, b, [], [], 5); %! assert (flag, 0); %! assert (x, ones (5, 1), -1e-6); %!test %! ## Maximum number of iteration reached %! A = hilb (100); %! b = sum (A, 2); %! [x, flag, relres, iter] = gmres (A, b, [], 1e-14); %! assert (flag, 1); %!test %! ## gmres recognizes that the preconditioner matrix is singular %! AA = 2 * eye (3); %! bb = ones (3, 1); %! I = eye (3); %! M = [1 0 0; 0 1 0; 0 0 0]; # the last row is zero %! [x, flag] = gmres (@(y) AA * y, bb, [], [], [], @(y) M \ y, @(y) y); %! assert (flag, 2); %!test %! A = rand (4); %! A = A' * A; %! [x, flag] = gmres (A, zeros (4, 1), [], [], [], [], [], ones (4, 1)); %! assert (x, zeros (4, 1)); %!test %! A = rand (4); %! b = zeros (4, 1); %! [x, flag, relres, iter] = gmres (A, b); %! assert (relres, 0); %!test %! A = toeplitz (sparse ([2, 1, 0, 0, 0]), sparse ([2, -1, 0, 0, 0])); %! b = A * ones (5, 1); %! [x, flag, relres, iter] = gmres (A, b, [], [], [], [], [], ... %! ones (5, 1) + 1e-8); %! assert (iter, [0, 0]); %!test %! A = rand (20); %! b = A * ones (20, 1); %! [x, flag, relres, iter, resvec] = gmres (A, b, [], [], 1); %! assert (iter, [1, 1]); %!test %! A = hilb (20); %! b = A * ones (20, 1); %! [x, flag, relres, iter, resvec] = gmres (A, b ,5, 1e-14); %! assert (iter, [4, 5]); %!test %! A = single (1); %! b = 1; %! [x, flag] = gmres (A, b); %! assert (class (x), "single"); %!test %! A = 1; %! b = single (1); %! [x, flag] = gmres (A, b); %! assert (class (x), "single"); %!test %! A = single (1); %! b = single (1); %! [x, flag] = gmres (A, b); %! assert (class (x), "single"); %!test %!function y = Afun (x) %! A = toeplitz ([2, 1, 0, 0], [2, -1, 0, 0]); %! y = A * x; %!endfunction %! [x, flag] = gmres ("Afun", [1; 2; 2; 3]); %! assert (x, ones (4, 1), 1e-6); %!test # preconditioned residual %! A = toeplitz (sparse ([2, 1, 0, 0, 0]), sparse ([2, -1, 0, 0, 0])); %! b = sum (A, 2); %! M = magic (5); %! [x, flag, relres] = gmres (A, b, [], [], 2, M); %! assert (relres, norm (M \ (b - A * x)) / norm (M \ b), 8 * eps);